Journal of Graph Algorithms and Applications
http://jgaa.info/ vol. 16, no. 3, pp. 651–673 (2012)
DOI: 10.7155/jgaa.00263
Inapproximability of Orthogonal Compaction
Michael J. Bannister David Eppstein Joseph A. Simons
Computer Science Department,
University of California, Irvine
Abstract
We show that several problems of compacting orthogonal graph drawings to use the minimum number of rows, area, length of longest edge or
total edge length cannot be approximated better than within a polynomial factor of optimal in polynomial time unless P = NP. We also provide
a fixed-parameter-tractable algorithm for testing whether a drawing can
be compacted to a small number of rows.
Submitted:
November 2011
Reviewed:
February 2012
Article type:
Regular paper
Revised:
Accepted:
Final:
February 2012 February 2012
March 2012
Published:
September 2012
Communicated by:
M. van Kreveld and B. Speckmann
This work was supported in part by NSF grant 0830403 and by the Office of Naval Research
under grant N00014-08-1-1015. A previous version of this paper appeared in the International
Symposium on Graph Drawing 2011.
E-mail addresses: mbannist@uci.edu (Michael J. Bannister) eppstein@uci.edu (David Eppstein)
jsimons@uci.edu (Joseph A. Simons)
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Introduction
Orthogonal graph drawing is a widely used graph drawing style for low-degree
graphs, in which each vertex is represented as a point or a rectangle in an integer
grid, and each edge is represented as a polyline composed out of axis-parallel line
segments [5]. When used for nonplanar graphs (Figure 1), orthogonal drawing
has several desirable properties including polynomial area, high angular resolution, and right-angled edge crossings; the last property, in particular, has been
shown to aid in legibility of graph drawings [9].
Typical orthogonal graph drawing systems employ a multiphase approach [1,
5] in which the input graph is planarized by replacing its crossings with vertices,
a topological embedding of the graph (specifying the ordering of the edges around
each vertex, but not the vertex and edge locations) is found, a flow algorithm
is used to orient the edges in a way that minimizes the number of bends [16],
and vertex coordinates are assigned. If vertices of degree greater than four
exist, they may be expanded to rectangles as another phase of this process [1].
Finally, the drawing is improved by compaction, a step in which the vertices
and bends of the graph are moved to new locations in order to reduce the area
of the drawing while preserving its edge orientations and other features.
Some positive algorithmic results are known for the final compaction step
for planar drawings; for instance, Bridgeman et al. [3] showed that planar orthogonal drawings in which the shapes of the faces in the drawing are restricted
(so-called turn-regular drawings) may be compacted into optimal area in polynomial time. In the case of variable size vertices, e.g. due to labels, Eiglsperger
et al. [6] have found heuristic based compaction algorithms that work well in
practice. Also, Klau et al. have shown that the compaction step can be done via
integer linear programming [12]. An experimental analysis of planar compaction
methods was done by Klau et al. [11]
However, when drawing nonplanar graphs, it may not be necessary or desirable for the compaction phase to preserve a fixed planarization of the graph. If
one is compacting one dimension of a drawing at a time, then for planar compaction it is only possible to map the rows of the drawing monotonically to a
smaller set of rows, while for nonplanar graphs it may also be useful to permute
the rows with respect to each other. This greater freedom to choose how to
Figure 1: Examples of non-planar orthogonal drawings.
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compact the drawing may lead to much greater savings in drawing area, but it
also leads to greater difficulty in finding a good compaction.
As Patrignani [15] showed, even for arbitrary planar orthogonal graph drawings, compacting the drawing in a way that minimizes its area, total edge length,
or maximum edge length is NP-hard. Although these results do not directly extend to the nonplanar case, NP-hardness in that case also follows from results
of Eades et al. on rectilinear (bendless) drawing [4], and Maňuch et al. where
certain restricted cases of rectilinear drawing are considered [13]. But since
compaction is performed primarily for aesthetic reasons (a smaller area drawing
allows the drawing to be viewed at a larger scale, making its features more legible), exact optimization may not be important if a layout with small area can
be achieved. Thus, we are led to the problem of how closely it is possible to approximate the minimum area layout. The problem of approximate compaction
for nonplanar orthogonal drawings was explicitly listed as open by Eiglsperger
et al. [5], and there appears to have been little progress on it since then.
In this paper we show that nonplanar compaction is hard even to approximate. There exists a real number c > 0 such that, unless P = NP, no polynomial
time algorithm can find a compaction of a drawing with n features that is within
a factor of nc of optimal. The main idea is to find approximation-preserving
reductions from graph coloring, a problem known to be hard to approximate.
We also give a fixed-parameter tractable algorithm for finding compactions that
use very small numbers of grid rows, for drawings in which such a compaction
is possible.
1.1
Variations of the compaction problem
In the compaction problems we study, the task is to move vertices and bends
while preserving the axis-parallel orientation (although not necessarily the direction) of each edge, to minimize the number of rows, area, longest edge, or
total edge length of the drawing. Our results apply either to orthogonal drawings
(drawings in which edges may be polylines with bends, possible for any graph of
maximum degree four) or rectilinear drawings (bendless drawings, only possible
for some graphs) [4, 7]; the distinction between bends and vertices is unimportant for our results.
We distinguish between three variants of the compaction problem, depending
on what vertex motions are allowed. In row-by-row compaction (Figure 2), the
compacted layout maps each row of the input layout to a row of the output; all
vertices that belong to the same row must move in tandem. In vertex-by-vertex
vertical compaction (Figure 3), each vertex or bend may move independently,
but only its y-coordinate may change; it must retain its horizontal position. In
vertex-by-vertex free compaction, vertices or bends may move arbitrarily in both
coordinate directions. In all three of these problems, edges or edge segments
must stay vertical or horizontal according to their orientation in the original
layout. The compaction is not allowed to cause any new intersection between
a vertex and a feature it was not already incident with, nor is it allowed to
cause any two edges or edge segments to overlap for nonzero length; however, it
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may introduce new crossings that were not previously present. Therefore, the
compacted drawing remains a valid representation of the graph; however, the
introduction of crossings may reduce the aesthetics and readability of the graph,
especially in cases where the initial input drawing is planar.
Figure 2: Example of row-by-row compaction.
Figure 3: Example of vertex-by-vertex compaction.
1.2
New results
We show the following results.
• In the row-by-row compaction problem, it is difficult to compact even
a drawing of a path graph (or a drawing of the two-vertex graph with
many bends). If the drawing has n vertices or bends, then for every
> 0 there is no polynomial time algorithm that can find a compacted
drawing whose number of rows, maximum edge length, total edge length,
or area is within O(n1/2− ) of optimal, unless P = NP. Moreover, even
finding drawings with a fixed number of rows is hard; it is NP-complete
to determine whether there exists a compaction with only three rows.
• In vertex-by-vertex vertical compaction, for every > 0, there is no polynomial time algorithm that can find a compacted drawing of graphs of
maximum degree three such that the number of rows, maximum edge
length, total edge length, or area of the drawing is within O(n1/3− ) of
optimal, unless P = NP. The same result also applies in the vertex-byvertex free compaction problem.
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• For the vertex-by-vertex or vertex-by-vertex free compaction problem of
three-dimensional orthogonal drawings, for every > 0, it is not possible
to approximate the minimum number of layers in any one dimension,
maximum edge length, total edge length, or volume to within O(n1/2− ) of
optimal in polynomial time, unless P = NP, nor is it possible in polynomial
time to determine whether a three-layer drawing exists.
• In vertex-by-vertex vertical compaction, there is an algorithm for testing
whether an orthogonal graph drawing can be compacted into k rows, whose
running time is O(k!n). Thus, the problem is fixed-parameter tractable.
• In vertex-by-vertex free vertical compaction, there is a simple linear time
algorithm for finding an optimal row compaction of a path.
• We provide a counterexample showing that an approximation algorithm
described in the conference version of this paper was incorrect.
2
Preliminaries
2.1
Orthogonal Drawing
We define an orthogonal drawing of a graph to be a drawing in which each vertex
is represented as a point in the Euclidean plane (although most of our results
apply as well to drawings in which the vertices are rectangles), and each edge
is represented as a polyline (a polygonal chain of line segments), with each line
segment parallel to one of the coordinate axes. (See, for example, Figure 1.)
If each edge is itself a line segment, the drawing is rectilinear ; otherwise, the
segments of a polyline meet at bends. Each vertex or bend must only intersect
the edges that it belongs to, and no two vertices or bends may coincide. Edges
may cross each other, but only at right angles, at points that are neither vertices
nor bends.
It is natural, in orthogonal graph drawing, to restrict the coordinates of the
vertices and bends to be integers. In this case, the width of a two-dimensional
drawing is the maximum difference between the x-coordinates of any two of its
vertices or bends, the height is the maximum difference between y-coordinates
of any two vertices or bends, and the area is the product of the width and
height.
A compaction of a drawing D is another drawing D0 of the same graph, in
which the vertices and bends of D0 correspond one-for-one with the vertices
and bends of D, and in which corresponding segments of the two drawings are
parallel to each other. Typically, D0 will have smaller height, width, or area
than D. We distinguish between three types of compaction:
• In row-by-row compaction, the x-coordinate of each vertex or bend remains
unchanged, and two vertices or bends that have the same y-coordinate in
D must continue to have the same y-coordinate in D0 (Figure 2).
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• In vertex-by-vertex vertical compaction, the x-coordinate of each vertex or
bend remains unchanged, but the y-coordinates may vary independently of
each other subject to the condition that the result remains a valid drawing
with edge segments parallel to the original drawing (Figure 3).
• In vertex-by-vertex free vertical compaction, the x- and y- coordinates of
each vertex or bend are free to vary independently of other vertices or
bends.
As can be seen in Figure 2, we allow compaction to introduce new edge crossings and to reverse the directions of edge segments. These concepts generalize
straightforwardly to three dimensions.
2.2
Graph Coloring and Inapproximability
In the graph coloring problem, we are given as input a graph and seek to color
the vertices of the graph with as few colors as possible, in such a way that the
endpoints of each edge are assigned different colors. Our results on the difficulty
of compaction are based on known inapproximability results for graph coloring,
one of the triumphs of the theory of probabilistically checkable proofs [8,10,17].
Theorem 1 (Zuckerman [17]) Let > 0 be any fixed constant. Then, unless
P = NP, there is no polynomial time algorithm that can compute the minimal
number of colors χ(G) in an optimal coloring of a given n-vertex graph within
a factor of n1− .
Our proofs use approximation-preserving reductions from coloring to compaction. Given a graph G to be colored, we will construct a different graph G0
and a drawing D of G0 such that the number of rows, maximum edge length, or
total edge length in a compaction D0 of D necessarily correspond to the number
of colors in a coloring of G. With a reduction of this type, the approximation
ratio for compacting D cannot be better than the approximation ratio for coloring G. However, D will in general have many more vertices and bends than
the number of vertices in G; the size of D will be at least proportional to the
number of edges in G, which is quadratic in its number of vertices. Therefore,
although the approximation ratio will remain unchanged as a number by our
reduction it will be expressed as a different function of the input size.
2.3
Notation
We write nG , nD , or (where unambiguous) n for the number of vertices in a
graph G or drawing D and mG , mD , or m for its number of edges. Additionally,
bD stands for the number of bends in drawing D. When we consider compaction
problems with the objectives of minimizing the number of rows, maximum edge
length, and total edge length in a drawing D we will use λ(D), µ(D), and τ (D)
to denote respectively the number of rows, maximum edge length, and total
edge length in an optimal compaction. χ(G) represents the chromatic number
of graph G.
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1
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1
2
3
3
5
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5
4
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Figure 4: Path constructed from a graph G with the walk 1, 5, 3, 2, 5, 4, 3, 5, 6.
3
Hardness of Row-By-Row Compaction
As a warm-up, we start with a simplified path compaction problem in which
every pair of objects on the same row of the drawing must move in tandem. Our
proof constructs a drawing of a path graph such that the valid row assignments
for our drawing are the same as the valid colorings of a given graph G.
Lemma 1 Given a graph G we can construct in polynomial time a rectilinear
drawing D of a path graph with O(mG ) vertices, such that λ(D) = χ(G).
Proof: Find a Chinese postman walk for G of length less than 2mG ; that is,
a walk that starts at an arbitrary vertex and visits each edge at least once,
allowing vertices to be visited multiple times. Such a walk may be found, for
instance, by doubling each edge of G and constructing an Euler tour of the
doubled graph. Let ui vi be the ith edge in the walk, where vi = ui+1 , and let
k ≤ 2mG be the number of edges in the walk. Additionally, choose arbitrary
distinct integer labels for the vertices of G with `(v) being the label for the
vertex v.
To construct the drawing D, for i from 0 to k, place vertices in the plane
at the points (i, `(ui )) and (i + 1, `(ui )), connected by a unit-length horizontal
edge. Additionally, for i from 0 to k − 1 draw a vertical edge from (i + 1, `(ui ))
to (i + 1, `(vi )). See Figure 4 for an example of such a construction.
Two rows in the drawing conflict if and only if the corresponding vertices in
G are adjacent. For every coloring of G, we may compact D by using one row
for the vertices of each color, and conversely for every row-by-row compaction
of D we may color G by using one color class for each row of the compaction
(Figure 5). Therefore, λ(D) = χ(G). Also, nD = 2k + 2 = O(mG ).
The same drawing D can equivalently be viewed as an orthogonal drawing
of the two-vertex graph K2 with O(mG ) bends. In the restricted model of compaction used in this section, horizontal compaction is disallowed, so optimizing
the area of a compaction of D is the same as optimizing the number of rows.
Theorem 2 Let > 0 be any positive fixed constant, and suppose that P 6= NP.
Then there does not exist a polynomial time algorithm that approximates the
number of layers or the area in an optimal row-by-row compaction of a given
orthogonal or rectilinear drawing D to within a factor of (nD + bD )1/2− .
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2
3
1
5
4
6
Figure 5: The rows of a compacted drawing D correspond to the colors in a
coloring of G.
Proof: Suppose for a contradiction that algorithm A can solve the row-by-row
1/2−
compaction problem to within a factor ρ ≤ nD
of optimal. Let A0 be an
algorithm for coloring an input graph G by performing the following steps:
1. Use Lemma 1 to construct a path drawing D from the given graph G.
2. Use algorithm A to compact D.
3. Color G using one color for each row of the compacted drawing.
Then the approximation ratio of algorithm A0 for coloring is the same number
ρ as the approximation ratio of algorithm A for compaction, whether measured
by area or by number of rows. However,
1/2−
ρ ≤ nD
1/2−
= O(mG
) = O(n1−2
),
G
an approximation ratio that contradicts Theorem 1.
The same reduction, together with the NP-completeness of graph 3-colorability, shows that it is NP-complete to determine whether a given drawing D
has a row-by-row compaction that uses at most three rows.
Theorem 3 The problem of determining if an orthogonal graph has a row-byrow compaction into at most three rows is NP-complete.
For the compaction problems with respect to longest edge, and total edge
length we will modify the construction in Lemma 1 by the addition of a frame
that will be used to measure χ(G) with τ (D) and µ(D). The frame is illustrated
in Figure 6.
Lemma 2 Given a graph G we can construct in polynomial time a rectilinear
drawing D of a path graph with O(mG + φ) vertices, such that
µ(D) = χ(G) + 1
and
φχ(G) ≤ τ (D) ≤ O(mG + φ)χ(G).
Proof: Given a graph G we construct D by first performing the construction
of Lemma 1, and then adding the vertices and edges as illustrated in Figure 6.
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φ
1
2
3
4
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Figure 6: We add a frame above and below the path construction of Figure 4
(shaded gray). The longest edge runs along the right side of the drawing from
the top frame to the bottom frame and will have length exactly χ + 1.
We claim that vertices created by Lemma 1 must stay between the two gray
regions in a row-by-row compaction. The leftmost vertex cannot be placed
above the upper gray region without causing an illegal edge overlap, and if one
vertex is placed above the upper gray region then all vertices must be placed
above the upper gray region to avoid illegal edge overlaps. Hence none of the
original vertices can be placed above the upper gray region (unless the entire
drawing is flipped). Similar reasoning shows that none of the original vertices
may be placed below the lower gray region.
By construction the φ edges to the right of the drawing are the longest edges
in the drawing, even after a row-by-row compaction. The length of one of these
edges is always 2 more than the length of the gap between the gray regions,
which is χ(G) − 1 by Lemma 1. Hence the length of the edges in an optimal
compaction is µ(D) = χ(G) + 1.
For the total edge length we note that there are at most 2mG + φ columns,
and each of these columns has at most 2χ(G) in length. The horizontal segments
make at most O(mG + φ) total length. So altogether we have that τ (D) is at
most O(mG + φ)χ(G). Considering only the vertical edges on the right we see
that φχ(G) ≤ τ (D), which gives the desired bounds.
Theorem 4 Let > 0 be any positive fixed constant, and suppose that P 6= NP.
Then there does not exist a polynomial time algorithm that approximates the
longest edge length or the total edge length in an optimal row-by-row compaction
of a given orthogonal or rectilinear drawing D to within a factor of (nD +
bD )1/2− .
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Proof: The proof of the longest edge case is the same as the proof of Theorem 2
for the number of rows where Lemma 2 is used in place of Lemma 1.
More work is needed for the total length case. Suppose for a contradiction
that an algorithm A could solve the row-by-row compaction problem with re1/2−
spect to total edge length to within a factor of nD
of optimal. Let A0 be
an approximation algorithm for χ(G) that compacts the drawing produced by
Lemma 2 with φ = n2G , and returns τ 0 (D)/n2G where τ 0 is the total edge length
of the approximate compaction.
From Lemma 2 we know that
n2G χ(G) ≤ τ (D) ≤ O(n2G )χ(G),
1/2−
and τ (D) ≤ τ 0 (D) ≤ nD
τ (D)
by assumption. Thus,
1/2−
χ(G) ≤ τ 0 (D)/n2G ≤ nD
1/2−
/n2G τ (D) ≤ O(n2G )/n2G nD
χ(G) ≤ O(n1−2
)χ(G)
G
contradicting Theorem 1.
4
Hardness of Vertex-By-Vertex Compaction
Our hardness result for vertex-by-vertex vertical compaction follows roughly
the same outline as Theorem 2: translate graph vertices into drawing features
such that two features can be compacted onto the same row if and only if the
corresponding graph vertices can be assigned the same color. However, if we
translated adjacencies in the graph to be colored into direct overlaps between
pairs of features, as we did for row-by-row compaction of path graphs, this
translation would only let us represent interval graphs, which are easily colored [14]. Instead we use an edge gadget depicted in Figure 7 to represent an
edge between two vertices of the input graph. This gadget has six vertices and
six line segments; the two vertices A and B of the gadget may be placed on two
line segments representing vertices of the input graph. This connection forces
the two line segments containing A and B to be placed on different rows of any
compacted drawing, even if these two line segments have no vertical overlap with
each other: one of the two line segments must be above the central rectangle of
the gadget, and the other must be below the central rectangle, although either
of these two orientations is possible.
B
A
Figure 7: The basic version of an edge gadget (left), connecting two horizontal
segments representing vertices (right).
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The use of these edge gadgets leads to a second difficulty in our reduction:
the number of rows in the compacted drawing will depend both on the features
coming from input graph vertices and the rows needed by the edge gadgets
themselves. In order to make the first of these two terms dominate the total, we
represent an input graph vertex by a vertex bundle of θ parallel line segments,
for some integer θ > 0. The edge gadgets may be modified to enforce that
all segments in one bundle be in different rows from all segments of a second
bundle, as shown in Figure 8, while only using a constant number of rows for
the gadget itself.
Figure 9 shows the complete reduction, for a graph G with five vertices and
six edges, and for θ = 1. Each vertex of G is represented as a horizontal black
line segment (or bundle of segments, for θ > 1), and each edge of G is represented
by an edge gadget. The vertices of G are numbered arbitrarily from 1 to nG ,
and these numbers are used to assign vertical positions to the corresponding
bundles of segments in the drawing. The edge gadgets are given x-coordinates
that allow them to attach to the two vertex bundles they should be attached
to, and y-coordinates that place them between these two vertex bundles.
Lemma 3 Given a graph G and a parameter θ we can construct in polynomial time an orthogonal drawing D such that the vertices of D have maximum
degree 3, nD = O(θnG + mG ), and
θχ(G) ≤ λ(D) ≤ θχ(G) + O(n2G ).
Proof: The construction of D is as described above. Since each edge gadget has
a constant number of vertices together they contribute O(mG ) to nD , and each
vertex bundle has O(θ) vertices together they contribute O(θnG ) to nD . Hence
nD = O(θnG + mG ). The bound on the degree is clear from the construction.
If G has a coloring with χ colors, it is possible to assign the vertex bundles
of D to χ sets of θ rows each, according to those colors, with an additional
O(nG ) rows between any two such sets to allow room for the edge gadgets
Figure 8: The full edge gadget for θ = 5.
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Figure 9: Example of the complete reduction for θ = 1.
1
3
2
5
4
3
2
4
5
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Figure 10: Example coloring based on the compaction of the drawing in Figure 9.
to be placed without interference with each other; see Figure 10. Therefore,
λ(D) ≤ θχ(G) + O(n2G ). If D0 is a compacted drawing of D, acyclically orient
the edges of G from the vertex whose bundle is below the edge gadget to the
vertex whose bundle is above the edge gadget, and assign each vertex v in G a
color indexed by the length of the longest path from a source to v in this acyclic
orientation. Then the number of colors needed equals the number of vertices
in the longest path, and the number of rows in D0 needed just for the vertices
in this path is θ times the number of vertices of G in the path. Therefore,
θχ(G) ≤ λ(D).
Theorem 5 If P 6= NP, then no polynomial time algorithm approximates the
number of rows or the area in an optimal vertex-by-vertex vertical compaction
1/3−
of a given orthogonal graph drawing to within a factor of nD
.
Proof: If an algorithm could achieve this approximation ratio for compaction,
we could get an O(n1−3
) ratio for coloring by applying Lemma 3 with θ = n2G ,
G
compacting the resulting drawing, and using the coloring derived from the compaction in the proof of Lemma 3, contradicting Theorem 1.
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Lemma 4 Given a graph G and parameters θ and φ we can construct in polynomial time an orthogonal drawing D such that the vertices of D have maximum
degree 3, nD = O(θnG + mG + φ), with the bounds
θχ(G) ≤ µ(D) ≤ θχ(G) + O(n2G )
when
θ > 6n2G
on the longest edge, and the bounds
τ (D) = Θ(n4G )χ(G)
when
θ, φ = Θ(n2G )
on the total edge length.
Proof: We add two rows of vertices one above and one below the drawing
produced by Lemma 3 (illustrated in Figure 9) by placing vertices above and
below the left most column of vertices in each vertex bundle. We also place
φ additional vertices on the top and the bottom to the right of the original
drawing. The vertices are then connected as illustrated by Figure 11.
First, we consider the bounds on the longest edge. For these bounds we let
φ = 1 and θ > 6n2 . The width of the original drawing (from Lemma 3) is 2nG
for the vertex bundle and 4mG for connecting the edge gadgets to the vertex
bundles; see the right side of Figure 8. So the width of the original drawing is
bounded above by 6n2G . When adding the additional frame (the green vertices
with purple edge in Figure 11), all added horizontal edges have length at most
2nG , more precisely twice the degree of the vertex being represented by the
bundle. Now, by construction the vertical edge on far right is the longest vertical
edge in the drawing, and has length at least θχ(G)+O(mG ) > 6n2G by Lemma 3,
making it the longest edge in the drawing even after compaction. So we have
that µ(D) = λ(D) − 1 for the constructed drawing, i.e., minimizing the longest
edge in the new drawing is equivalent minimizing the number of rows in the
original drawing. The bounds for longest edge now follow from the bounds in
Lemma 3.
For the bounds on total edge length, we let θ, φ = Θ(n2G ). Each vertex bundle
contributes O(θnG ) for a total of O(θn2G ). The height of each edge gadget is
O(θχ(G)) from the vertex bundle plus O(mG ) from the other edge gadgets. The
horizontal contribution from each edge gadgets is O(n2G ), as the width of the
drawing is at most 6n2G . Altogether each edge gadget contributes O(θχ(G)+n2G )
for a total of O(n2G θχ(G) + n4G ). The added frame contributes O(n2G + φ) in
horizontal edge length, and φθχ(G) in vertical edge length. Since φ = Θ(n2G )
the vertical lines holding the the drawing in the frame do not contribute to the
asymptotic complexity. Summing up the total edge length,
τ (D) = O(θn2G ) + O(n2G θχ(G) + n4G ) + φθχ(G) = Θ(n4G )χ(G),
we get the desired bounds.
Theorem 6 If P 6= NP, then no polynomial time algorithm approximates the
length of the longest edge or total edge length in an optimal vertex-by-vertex
vertical compaction of a given orthogonal graph drawing to within a factor of
1/3−
nD
.
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φ
Figure 11: Long-edges-frame added to the drawing in Figure 9
Proof: If an algorithm could achieve this approximation ratio for compaction,
we could get an O(n1−3 ) ratio for coloring by applying Lemma 4 with θ = 7n2
and φ = 1 and compacting with respect to longest edge, or by applying Lemma 4
with θ = φ = n2 and compacting with respect to total edge length.
5
Hardness of Vertex-By-Vertex
Free Compaction
In the reduction from the previous section, allowing the vertices to move horizontally as well as vertically does not make any difference in how much vertical
compaction is possible. However, if we want to prove inapproximability for
minimal-area compaction, we also need to worry about horizontal compaction.
By making the width uncompactable we may make the vertical compaction
factor the same as the area compaction factor.
Lemma 5 From a drawing D a drawing D0 can be constructed by adding at
most O(nD ) vertices, such that λ(D0 ) = λ(D) + 1 and D0 is uncompactable in
the horizontal direction. If D has maximum degree three, then so does D0 .
Proof: Place a line of vertices on a new row below D; for each set of vertices
with a given x-coordinate in D, add a vertex on the new row at the same xcoordinate. Connect the added vertices with horizontal edges, and add a vertical
edge to connect these vertices to D at the point of D that is rightmost on its
bottom row, as shown in Figure 12. This added layer conflicts with all existing
horizontal layers, and forces D0 to be uncompactable in the horizontal direction.
Theorem 7 Unless P = NP, it is impossible to find vertex-by-vertex free compactions with respect to number of rows, area, longest edge length or total edge
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Figure 12: Adding a row of vertices to D prevents horizontal compaction.
1/3−
length within a factor of nD
of optimal in polynomial time.
Proof: For the case of compaction with respect to area we apply Lemma 5
to reduce compaction with respect to area to compaction with respect to the
number of rows. Since we only added a linear number of vertices the approximation ratios are the same. The rest of the theorem follows from the results on
vertex-by-vertex vertical compaction.
6
Hardness of Three-Dimensional Compaction
Our hardness result for three-dimensional vertical compaction (along the z-axis)
follows from the construction of a drawing whose valid two-dimensional layer
assignments are the same as the valid colorings of a graph G. We assign to each
vertex in G a horizontal layer parallel to the xy-plane containing an L-shaped
pair of line segments, such that when projected vertically onto the xy-plane
every two of these L-shaped gadgets cross each other. For each edge in G
we place a vertical edge in the drawing connecting the L-shaped gadgets that
correspond to the endpoints of the edge. Figure 13 shows an example.
Lemma 6 Given a graph G we can construct in polynomial time a 3D orthogonal drawing D with maximum degree three such that nD = 3nG +2mG = O(n2G ),
and such that the number of layers in an optimal y-compaction is χ(G).
Proof: Our construction uses three vertices in D to form the L-shaped gadget
for each vertex in G, and two more vertices for the vertical edge corresponding
to each edge in G, so nD = 3nG + 2mG = O(n2G ). By construction the only
conflicting features in a vertical (along the z-axis) compaction come from the
vertical edges, forcing χ(G) to be the number of layers in an optimal compaction
of D.
Theorem 8 If P 6= NP, then there does not exist a polynomial time algorithm
that approximates the number of layers in an optimal vertex-by-vertex or vertexby-vertex free vertical compaction of a given three dimensional orthogonal draw1/2−
ing to within a factor of nD
.
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Bannister, Eppstein, Simons Compaction for Graph Drawings
Proof: In this case nD = O(n2G ), and the number of layers in an optimal
compaction is equal to χ(G). The proof follows same lines as the proofs in
Section 3 on row-by-row-compaction.
As we did in previous sections we will reduce compaction with respect to
volume, longest edge, and total edge length to compaction with respect to layering via a frame. In this case we will place a mesh layer above and below the
drawing, keeping the original drawing between the two layers.
Lemma 7 Given a graph G with more than 4 vertices we can construct in
polynomial time a 3D orthogonal drawing D with maximum degree three with
nD = O(n2G ), and such that
µv (D) = χ(G) + 1
and
n2G χ(G) ≤ τ (D) ≤ O(n2G )χ(G)
where µv (D) is the length of the longest vertical edge in an optimal vertical
compaction.
4
3
2
2
3
1
1
4
Figure 13: Reduction from coloring to three-dimensional compaction where y is
the vertical direction.
2n
Figure 14: Left: View of the 3D-drawing in Figure 13 from above. Vertical
edges meet at lighter shaded vertices. Right: Top/bottom frame.
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Proof: From the drawing produced by Lemma 6 we add O(n2G ) edges, and
consequently O(n2G ) vertices, in 2nG parallel paths above and below the drawing
as illustrated in Figure 14, where the frame on the right side of the drawing is
placed directly over and below the 3D drawing to the left. The diagonal nodes
are connected to the ends of the L-gadgets from the top and the bottom to keep
them between the two frames. The sub-diagonal green vertices are connected
with edges between the upper and lower frame. This construction is illustrated
in Figure 15.
By construction the longest vertical edges are the edges connecting the top
and bottom frames; their length is one unit more than the number of layers
between the top and the bottom frame. Hence µv (D) = χ(G) + 1 by Lemma 6.
The number of edges connecting the two layers is given by
1 + 2 + 3 + · · · + (2nG − 2) = O(n2G )
and is greater than n2G for n > 4. The total edge length used by the L-gadgets
is at most 2n2G , and χ(G)mG for the connecting segments. The top and bottom frames add O(n2G ) length to the drawing, and the connecting edges add
O(n2G )χ(G). So we have that n2G χ(G) ≤ τ (D) ≤ O(n2G )χ(G).
Figure 15: 3D frame construction
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Bannister, Eppstein, Simons Compaction for Graph Drawings
Theorem 9 If P 6= NP, then there does not exist a polynomial time algorithm
that approximates the volume, longest vertical edge, or total edge length in an
optimal layer compaction of a given three dimensional orthogonal drawing to
1/2−
within a factor of nD
.
Proof: Having the bounds in Lemma 7 the proof follows as before.
We can also achieve a similar result for longest edge, without having to limit
our result to the longest vertical edge, by replacing each L-shaped vertex gadget
by an L-bundle, similar to our vertex bundle in the two dimensional case. This
is needed to prevent the longest edge from being a nG length leg of the L-gadget.
We omit the details.
7
Coloring Counterexample
In the conference version of this paper we included an approximation algorithm
for compaction which was later discovered to be flawed. In this algorithm we
defined a incompatibility graph, where the vertices of the incompatibility graph
represent sets of features in the drawing that must move in tandem. Two vertices
of the incompatibility graph are connected by an edge when the drawing features
they represent cannot be compacted to the same row of the drawing, i.e., when
they contain parts of the drawing that are directly above one another. A coloring
of the incompatibility graph was then used to to compact the drawing, using a
row for each color. The problem is that although compacting a drawing does
produce a coloring of its incompatibility graph, the converse does not hold. In
Figure 16 we see a drawing whose incompatibility graph is 2-colorable (in fact
it is a path), but which is completely uncompactable.
Figure 16: An uncompactable graph with a 2-colorable incompatibility graph.
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Fixed-Parameter Tractability of
Vertex-By-Vertex Vertical Compaction
Lemma 8 Given an orthogonal drawing D we can compact D into k rows in
O(k!(nD + bD )) time, if such a compaction is possible.
Proof: We construct local assignments of the features into k rows via a leftto-right plane sweep. The input drawing may be assumed to be given with
coordinates on an n × n grid, so its features can be sorted in linear time using
bucket sorting. While sweeping the plane we maintain the set of input features
that intersect the sweep line along with a record of valid assignments of these
features into the k rows of our output drawing.
When a feature first intersects the sweep line we try to place it into the
collection of valid assignments. If there are ` features intersecting the sweep
line prior to the insertion, we have at most `! k` valid assignments to consider.
In each of these valid assignments there are k − ` free rows. Altogether at most
k! configurations will be considered for each feature insertion. When the sweep
line moves past a feature its row is freed for future use.
If at any point we cannot find any valid assignment for a new feature, we
conclude that a compaction into k rows is not possible. On the other hand if
the last feature can be placed into a valid assignment, then a compaction into
k layers is possible. To recover the global assignment of horizontal features into
rows, we may backtrack through the sets of local assignments as is standard for
a dynamic programming algorithm of this type.
Theorem 10 An optimal vertex-by-vertex vertical compaction of an orthogonal
drawing D can be found in O(λ!(nD + bD )) time where λ = λ(D).
Proof: Apply Lemma 8 for k = 1, 2, 3, . . . until finding a value of k for which
a valid layering exists.
9
Vertex-By-Vertex Free Compaction for Paths
In contrast with row-by-row compaction (Section 3), we have a polynomial
time algorithm for vertex-by-vertex free compaction of a path. In fact, in the
following theorem, we give a simple, linear time algorithm to achieve an optimal
compaction. In this section by a vertical subpath we mean a maximal subpath
containing only vertical edges, similarly for a horizontal subpath
Theorem 11 An optimal vertex-by-vertex free row compaction for an orthogonal path can by found in O(nD + bD ) time. Such a compaction will be into k
rows where k is the number of vertices on the longest vertical path.
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Bannister, Eppstein, Simons Compaction for Graph Drawings
Figure 17: Compacting a path into k = 5 rows. Here the colors on vertices are
simply meant to help clarify the correspondence between the two paths.
Proof: First we replace the bends in the path with special “bend vertices” that
will be changed back to bends after the compaction. Now, the number of rows
required by a compacted drawing must be at least the number of vertices in the
longest vertical subpath, since each of these vertices must be placed in its own
row to achieve an unambiguous drawing. We will show that we can compact
the drawing into exactly this number of rows, and therefore optimally compact
the drawing. To perform a vertex-by-vertex free compaction of an orthogonal
path we perform the following steps, which are illustrated in Figure 17.
1. Compute the number k of vertices is the longest vertical subpath by walking the path.
2. Order the horizontal subpaths in walk order. Alternate placing horizontal
paths, in path order, in rows 1 and k overlapping in their endpoints.
3. Connect the overlapping horizontal paths with vertical lines, and place
the appropriate number of vertices on the edge to construct the vertical
path between the overlapping horizontal paths.
The end result is an optimally vertex-by-vertex free compacted path with respect
to the number of rows.
In this algorithm moving vertices horizontally was essential, so such a compaction is not possible in vertex-by-vertex vertical compaction. In contrast to
our result, Brandes et al. showed that the problem of orthogonal order preserving drawing is NP-hard for a path [2]. We leave as an interesting open problem
the complexity of vertex-by-vertex vertical compaction for a path.
JGAA, 16(3) 651–673 (2012)
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671
Conclusions
Our investigations have determined lower bounds for several different approximation and fixed-parameter versions of the compaction problem. Currently we
have no upper bounds on the √
approximation ratio for these compaction problems. With our techniques O( n) is a natural upper bound, as our reductions
produce drawings with at least one vertex for each edge in the original graph,
approximately squaring the number of vertices in the problem. Can algorithms
be constructed to match this expected upper bound? Can the lower bounds of
Ω(n1/3− ) be improved to Ω(n1/2− ) or higher? We leave these questions open
for future research.
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Bannister, Eppstein, Simons Compaction for Graph Drawings
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